联合零范数稀疏和时序差分低秩的运动捕获数据恢复算法

doi:10.11996/JG.j.2095-302X.2026030616

图学学报 ›› 2026, Vol. 47 ›› Issue (3): 616-628.DOI: 10.11996/JG.j.2095-302X.2026030616

• 计算机图形学与虚拟现实 • 上一篇下一篇

联合零范数稀疏和时序差分低秩的运动捕获数据恢复算法

胡文玉¹^,², 徐浩¹, 邱熙雯¹, 易云¹^,²()

¹ 赣南师范大学数学与计算机科学学院，江西赣州 341000
² 江西省教育厅数据科学与人工智能重点实验室，江西赣州 341000

收稿日期:2025-11-03 接受日期:2026-03-27 出版日期:2026-06-30 发布日期:2026-06-30
通讯作者:易云，E-mail：yiyun@gnnu.edu.cn
基金资助:
国家自然科学基金(62266002);国家自然科学基金(62362003);江西省自然科学基金(20252BAC250007);赣南师范大学研究生创新基金项目(YCXJ24-A12)

Motion capture data recovery by combining zero-norm sparsity and temporal difference low rank

HU Wenyu¹^,², XU Hao¹, QIU Xiwen¹, YI Yun¹^,²()

¹ School of Mathematics and Computer Science, Gannan Normal University, Ganzhou Jiangxi 341000, China
² Key Laboratory of Data Science and Artificial Intelligence of Jiangxi Education Institutes, Ganzhou Jiangxi 341000, China

Received:2025-11-03 Accepted:2026-03-27 Published:2026-06-30 Online:2026-06-30
Contact: YI Yun，E-mail：yiyun@gnnu.edu.cn
Supported by:
National Natural Science Foundation of China(62266002);National Natural Science Foundation of China(62362003);Natural Science Foundation of Jiangxi Province(20252BAC250007);Postgraduate Innovation Fund Project of Gannan Normal University(YCXJ24-A12)

摘要/Abstract

摘要：

针对运动捕获数据在采集与传输过程中普遍存在的噪声干扰以及标记点缺失问题，提出一种联合 ${l}_{0}$范数稀疏和时序差分低秩极小化的运动捕获数据恢复模型 (ZTDL)。首先，引入时序差分的低秩正则项以刻画运动捕获数据的全局低秩性与时序光滑性，同时结合 ${l}_{0}$范数和Frobenius范数分别表征稀疏缺失噪声和高斯叠加噪声。其次，利用 ${l}_{0}$范数邻近算子的特殊性质将非凸恢复模型等价地转换为含有二值掩码矩阵的双变量优化问题进行求解，所得模型能够同时精确估计缺失区域与恢复运动捕获数据。随后，采用交替方向乘子法和(逆)离散余弦变换对模型进行快速求解。在理论上，严格证明了算法能够依坐标收敛于局部极小解。最后，在CMU数据集和HDM05数据集上，将ZTDL算法与多种经典算法，如TSMC、TRNN、IRNN-Lp、TSPN算法与深度学习算法等，进行大量的数值对比实验。恢复误差和视觉效果比较结果表明，ZTDL无论是估计缺失区域还是恢复失真的运动捕获数据都具有显著的优越性。

关键词: 运动捕获数据, 时序差分, 稀疏低秩, 非凸优化, 零范数

Abstract:

To address the prevalent noise interference and missing-marker problem during the collection and transmission of Motion Capture (MoCap) data, a recovery model combining the Zero-norm sparsity and Temporal Differential Low-rank minimization (ZTDL) was proposed. Firstly, a temporal difference low-rank regularization term was introduced to capture the global low-rank property and temporal smoothness of MoCap data. Besides, the l₀ norm and the Frobenius norm were employed to characterize sparse missing noise and additive Gaussian noise. Secondly, the non-convex recovery model was transformed into the optimization problem involving a binary mask matrix by exploiting the properties of the l₀ norm. This model enabled the simultaneous estimation of missing regions and the restoration of MoCap data. The optimization problem was efficiently solved using the Alternating Direction Method of Multipliers (ADMM) framework sand the (inverse) Discrete Cosine Transform (DCT/IDCT). The algorithm was rigorously proven to converge to a local minimum in a coordinate-wise manner in theory. Finally, extensive comparative experiments were conducted on the benchmark CMU and HDM05 datasets. The ZTDL algorithm was evaluated against a range of classical methods such as TSMC, TRNN, IRNN-Lp, and TSPN, as well as deep learning approaches. The restoration results, including recovery error and visual effect, demonstrated the significant superiority of ZTDL in both missing-region estimation and corrupted-data restoration.

Key words: motion capture data, temporal difference, sparse and low-rank, non-convex optimization, zero-norm

中图分类号:

TP391.41

胡文玉, 徐浩, 邱熙雯, 易云. 联合零范数稀疏和时序差分低秩的运动捕获数据恢复算法[J]. 图学学报, 2026, 47(3): 616-628.

HU Wenyu, XU Hao, QIU Xiwen, YI Yun. Motion capture data recovery by combining zero-norm sparsity and temporal difference low rank[J]. Journal of Graphics, 2026, 47(3): 616-628.

图/表 16

图1 运动捕获数据在时序差分前后的奇异值变化曲线图比较((a) 跑步; (b) 体操)

Fig. 1 The comparison of singular value curves before and after temporal difference for two motion capture data ((a) Run; (b) Gymnastics)

图2 随机噪声数据的RMSE折线图((a) 行走; (b) 跑步; (c) 体操; (d) 跳舞)

Fig. 2 RMSE curve of random noisy data ((a) Walk; (b) Run; (c) Gymnastics; (d) Dance)

图3 随机缺失数据的RMSE折线图((a) 行走; (b) 跑步; (c) 体操; (d) 跳舞)

Fig. 3 RMSE curve of randomly missing data ((a) Walk; (b) Run; (c) Gymnastics; (d) Dance)

图4 连续缺失数据的RMSE折线图((a) 行走; (b) 跑步; (c) 体操; (d) 跳舞)

Fig. 4 RMSE curve of continuously missing data ((a) Walk; (b) Run; (c) Gymnastics; (d) Dance)

图5 混合污染数据的RMSE折线图((a) 行走; (b) 跑步; (c) 体操; (d) 跳舞)

Fig. 5 RMSE curve of mixed polluted data ((a) Walk; (b) Run; (c) Gymnastics; (d) Dance)

表1 各算法恢复Walk序列的MRMSE比较

Table 1 MRMSE comparisons of various algorithms in recovering the Walk sequence

算法	随机噪声	随机缺失	连续缺失	混合污染
TSMC	0.198 4	0.180 2	0.057 1	0.234 2
TrNN	0.165 6	0.085 9	0.049 7	0.140 4
IRNN-LP	0.163 7	0.066 7	0.049 8	0.136 1
TSPN	0.138 1	0.079 5	0.062 4	0.135 5
LNM-QR	0.137 7	0.075 7		0.136 9
MFF-N	0.119 9	0.066 3	0.035 3	0.135 6
ZTDL	0.119 1	0.066 3	0.034 5	0.123 7

表2 各算法恢复Run序列的MRMSE比较

Table 2 MRMSE comparisons of various algorithms in recovering the Run sequence

算法	随机噪声	随机缺失	连续缺失	混合污染
TSMC	0.585 0	0.469 8	0.177 6	0.427 9
TrNN	0.453 6	0.352 6	0.168 1	0.415 0
IRNN-LP	0.453 5	0.232 5	0.171 2	0.330 5
TSPN	0.343 5	0.277 6	0.178 1	0.367 6
LNM-QR	0.340 1	0.227 4		0.274 9
MFF-N	0.249 3	0.219 9	0.085 7	0.190 2
ZTDL	0.248 7	0.219 1	0.075 1	0.185 5

表3 各算法恢复Gymnastics序列的MRMSE比较

Table 3 MRMSE comparisons of various algorithms in recovering the Gymnastics sequence

算法	随机噪声	随机缺失	连续缺失	混合污染
TSMC	0.377 0	0.322 6	0.162 7	0.320 3
TrNN	0.298 6	0.328 2	0.147 9	0.355 0
IRNN-LP	0.298 7	0.157 6	0.145 3	0.230 8
TSPN	0.231 5	0.154 2	0.166 9	0.231 4
LNM-QR	0.234 1	0.156 8		0.197 4
MFF-N	0.181 5	0.141 3	0.081 6	0.148 4
ZTDL	0.181 5	0.141 2	0.071 1	0.147 6

表4 各算法恢复Dance序列的MRMSE比较

Table 4 MRMSE comparisons of various algorithms in recovering the Dance sequence

算法	随机噪声	随机缺失	连续缺失	混合污染
TSMC	0.218 4	0.178 2	0.088 7	0.214 7
TrNN	0.183 2	0.236 5	0.082 8	0.256 5
IRNN-LP	0.183 3	0.118 8	0.080 6	0.188 4
TSPN	0.153 6	0.083 1	0.099 3	0.147 8
LNM-QR	0.158 1	0.091 8		0.144 9
MFF-N	0.132 3	0.075 7	0.050 9	0.134 6
ZTDL	0.132 1	0.075 6	0.041 4	0.123 6

图6 7种算法恢复4种30%缺失污染运动序列的视觉效果比较((a) 行走; (b) 跑步; (c) 体操; (d) 跳舞)

Fig. 6 Visual comparisons of seven Algorithms in recovering the 4 sequences with 30% missing data ((a) Walk; (b) Run; (c) Gymnastics; (d) Dance)

图7 7种算法恢复4种30%噪声污染运动序列的视觉效果比较((a) 行走; (b) 跑步; (c) 体操; (d) 跳舞)

Fig. 7 Visual comparisons of seven Algorithms in recovering four sequences with 30% noisy data ((a) Walk; (b) Run; (c) Gymnastics; (d) Dance)

图8 本文方法与其他4种经典算法在不同污染情况下的MRMSE直方图比较((a) 随机噪声; (b) 随机缺失; (c) 连续缺失; (d) 混合噪声)

Fig. 8 MRMSE bar chart comparisons for this paper method and four other classical algorithms under different pollution conditions ((a) Random noise; (b) Randomly missing; (c) Continuously missing; (d) Mixed noise)

图9 LSTM[27]与ZTDL的RMSE折线图比较 ((a) 随机噪声；(b) 随机缺失)

Fig. 9 RMSE curves comparisons of the LSTM[25] and ZTDL methods ((a)Random noise; (b) Randomly missing)

表5 Transformer[25]与ZTDL的MRMSE比较

Table 5 MRMSE comparisons between the Transformer [25] and ZTDL methods

算法	缺失比例/%	Basketball	Boxing	Jump
Transformer^[25]	10	0.44	2.14	0.52
ZTDL	10	0.73	0.59	0.41
Transformer^[25]	20	0.57	2.63	0.56
ZTDL	20	0.76	0.63	0.45
Transformer^[25]	30	0.59	2.56	0.59
ZTDL	30	0.82	1.35	0.48

图10 TS和ZTDL在随机缺失污染估计缺失区域的ACC条形图

Fig. 10 ACC bar chart of TS and ZTDL in the missing region estimation in the case of randomly missing pollution

表6 随机缺失下各算法在不同ACC值时恢复Gymnastics的MRMSE比较

Table 6 MRMSE comparisons of various algorithms with different ACC values for recovering the randomly missing Gymnastics sequence

算法	ACC
算法	0.889 0	0.997 8	1
TSMC	10.701 7	0.032 6	0.032 5
TRNN	9.603 0	0.028 2	0.027 9
TSPN	10.205 9	0.025 8	0.025 8
IRNN-Lp	6.285 2	0.026 2	0.026 1
ZTDL	10.935 7	0.015 0	0.014 8

参考文献 29

[1]	CHEN K, WANG Y P, ZHANG S H, et al. MoCap-solver: a neural solver for optical motion capture data[J]. ACM Transactions on Graphics, 2021, 40(4): 84.
[2]	DONG J T, SHUI Q, ZHANG Y Q, et al. Motion capture from internet videos[C]//The 16th European Conference on Computer Vision-ECCV 2020. Cham: Springer, 2020: 210-227.
[3]	XIA G Y, SUN H J, ZHANG G Q, et al. Human motion recovery jointly utilizing statistical and kinematic information[J]. Information Sciences, 2016, 339: 189-205. DOI URL
[4]	FENG Y F, JI M M, XIAO J, et al. Mining spatial-temporal patterns and structural sparsity for human motion data denoising[J]. IEEE Transactions on Cybernetics, 2015, 45(12): 2693-2706. DOI PMID
[5]	MARTINI E, CALANCA A, BOMBIERI N. Denoising and completion filters for human motion software: a survey with code[J]. Computer Science Review, 2025, 58: 100780. DOI URL
[6]	MOHAOUI S, DMYTRYSHYN A. Low-rank completion for motion capture data recovery: approaches, constraints, and algorithms[J]. Computer Science Review, 2026, 60: 100878. DOI URL
[7]	CANDÈS E, RECHT B. Exact matrix completion via convex optimization[J]. Communications of the ACM, 2012, 55(6): 111-119.
[8]	CANDES E J, TAO T. The power of convex relaxation: near-optimal matrix completion[J]. IEEE Transactions on Information Theory, 2010, 56(5): 2053-2080. DOI URL
[9]	LAI R Y Q, YUEN P C, LEE K K W. Motion capture data completion and denoising by singular value thresholding[EB/OL]. [2025-07-02]. https://dblp.uni-trier.de/db/conf/eurographics/eg-short2011.html#conf/eurographics/LaiYL11.
[10]	FENG Y F, XIAO J, ZHUANG Y T, et al. Exploiting temporal stability and low-rank structure for motion capture data refinement[J]. Information Sciences, 2014, 277: 777-793. DOI URL
[11]	赫高峰, 彭淑娟, 柳欣. 结合模糊聚类和投影近似点算法的缺失人体运动捕捉数据重构[J]. 计算机辅助设计与图形学学报, 2015, 27(8): 1416-1425.
	HE G F, PENG S J, LIU X, et al. Missing human motion capture data recovery via fuzzy clustering and projected proximal point algorithm[J]. Journal of Computer-Aided Design & Computer Graphics, 2015, 27(8): 1416-1425(in Chinese).
[12]	胡文玉, 朱雪芳, 易云. 利用Capped核范数正则化的人体运动捕获数据恢复[J]. 计算机辅助设计与图形学学报, 2023, 35(8): 1184-1196.
	HU W Y, ZHU X F, YI Y. Human motion capture data recovery using capped nuclear norm regularization[J]. Journal of Computer-Aided Design & Computer Graphics, 2023, 35(8): 1184-1196(in Chinese).
[13]	LU C Y, TANG J H, YAN S C, et al. Nonconvex nonsmooth low rank minimization via iteratively reweighted nuclear norm[J]. IEEE Transactions on Image Processing, 2016, 25(2): 829-839. DOI PMID
[14]	HU W Y, WANG Z, LIU S, et al. Motion capture data completion via truncated nuclear norm regularization[J]. IEEE Signal Processing Letters, 2018, 25(2): 258-262. DOI URL
[15]	CHEN B J, SUN H J, XIA G Y, et al. Human motion recovery utilizing truncated schatten p-norm and kinematic constraints[J]. Information Sciences, 2018, 450: 89-108. DOI URL
[16]	RAJ M S S, GEORGE S N. A fast and efficient approach for human action recovery from corrupted 3-D motion capture data using QR decomposition-based approximate SVD[J]. IEEE Transactions on Human-Machine Systems, 2024, 54(4): 395-405. DOI URL
[17]	HUANG H Y, LI Q M, GAO Y, et al. Multi-level fine-grained fusion based robust low-rank approximation for human motion data restoration[J]. Neurocomputing, 2025, 658: 131587. DOI URL
[18]	YAN M. Restoration of images corrupted by impulse noise and mixed Gaussian impulse noise using blind inpainting[J]. SIAM Journal on Imaging Sciences, 2013, 6(3): 1227-1245. DOI URL
[19]	SUN L, JEON B, ZHENG Y H, et al. Hyperspectral image restoration using low-rank representation on spectral difference image[J]. IEEE Geoscience and Remote Sensing Letters, 2017, 14(7): 1151-1155. DOI URL
[20]	BECK A. First-order methods in optimization[M]. Philadelphia: SIAM, 2017: 129-178.
[21]	BOYD S, PARIKH N, CHU E, et al. Distributed optimization and statistical learning via the alternating direction method of multipliers[J]. Foundations and Trends^® in Machine Learning, 2011, 3(1): 1-122.
[22]	CAI J F, CANDÈS E J, SHEN Z W. A singular value thresholding algorithm for matrix completion[J]. SIAM Journal on Optimization, 2010, 20(4): 1956-1982. DOI URL
[23]	Carnegie Mellon University. Motion capture database[EB/OL]. [2025-03-16]. http://mocap.cs.cmu.edu.
[24]	Max Planck Institute for Informatics. Motion capture database hdm05[EB/OL]. [2025-03-16]. http://resources.Mpi-inf.mgp.de/HDM05/.
[25]	ZHANG J J, PENG J L, LV N. Spatial-temporal transformer network for human Mocap data recovery[C]// 2023 IEEE International Conference on Image Processing. New York: IEEE Press, 2023: 2305-2309.
[26]	HOLDEN D. Robust solving of optical motion capture data by denoising[J]. ACM Transactions on Graphics (TOG), 2018, 37(4): 165.
[27]	PAN X Y, ZHENG B W, JIANG X W, et al. A locality-based neural solver for optical motion capture[C]// SIGGRAPH Asia 2023 Conference Papers. New York: ACM, 2023: 117.
[28]	YOU Y X, LIU H, LI X, et al. Gator: graph-aware transformer with motion-disentangled regression for human mesh recovery from a 2D pose[C]// ICASSP 2023-2023 IEEE International Conference on Acoustics, Speech and Signal Processing. New York: IEEE Press, 2023: 1-5.
[29]	MOHAOUI S, DMYTRYSHYN A. CP decomposition-based algorithms for completion problem of motion capture data[J]. Pattern Analysis and Applications, 2024, 27(4): 133. DOI

联合零范数稀疏和时序差分低秩的运动捕获数据恢复算法

Motion capture data recovery by combining zero-norm sparsity and temporal difference low rank

RichHTML

PDF (PC)

可视化

摘要/Abstract

引用本文

使用本文

图/表 16

参考文献 29

相关文章 15

编辑推荐

Metrics

本文评价

[1]	周雪杨, 沈旭昆, 胡勇. 基于神经场逆渲染的三维重建研究综述[J]. 图学学报, 2026, 47(3): 449-471.
[2]	徐晓峰, 徐延宁, 王璐. 控制变量技术在渲染领域中的应用综述[J]. 图学学报, 2026, 47(3): 472-491.
[3]	王悦凝, 才让当知, 昝嵘, 孟磊. 基于轻量级网络的藏文复杂背景文字识别方法研究[J]. 图学学报, 2026, 47(3): 492-499.
[4]	李霁潼, 何金旭, 薛苏玲, 张俊, 娄路. 基于VGGT与显著性引导体素化的高效3DGS[J]. 图学学报, 2026, 47(3): 500-510.
[5]	李秀梅, 周正鑫, 孙军梅. 一种定位分支辅助的多任务协同图像伪造检测模型[J]. 图学学报, 2026, 47(3): 524-533.
[6]	唐海英, 李芳. 基于多模态Beat-STMAN网络模型的舞蹈动作识别方法[J]. 图学学报, 2026, 47(3): 534-542.
[7]	廖健康, 张严辞. 针对过度重建问题的频率强度高斯泼溅算法[J]. 图学学报, 2026, 47(3): 564-575.
[8]	张益, 王振, 刘艳丽, 邢冠宇. 渐进式时空细节增强视频去雾算法[J]. 图学学报, 2026, 47(3): 576-588.
[9]	卢德辉, 宋琢, 黄志超, 田时雨, 李慧敏, 田茂, 邓逸川. 基于TrueSkill排序与深度学习的绿色工地主观视觉感知预测[J]. 图学学报, 2026, 47(3): 641-652.
[10]	赵敏, 王妞娜, 严潼颖, 朱凌建. 基于L型靶标的相机标定及文物二维数字化[J]. 图学学报, 2026, 47(2): 423-431.
[11]	陈梦琪, 赵俊莉, 邓晓丹. 基于大模型的皮肤病图像掩膜生成与分割[J]. 图学学报, 2026, 47(2): 322-331.
[12]	向婷, 唐卓, 郑佳丽, 陈长建, 吕斐, 李肯立. 基于生成模型的图像数据增强方法综述[J]. 图学学报, 2026, 47(2): 235-250.
[13]	周腾龙, 杨文杰, 阴绍桦, 于元隆. 基于颜色多粒度学习的文本-图像行人再识别[J]. 图学学报, 2026, 47(2): 275-285.
[14]	房友江, 王世豪, 张亮, 段可然, 刘越, 魏小鹏, 杨鑫. 基于图拓扑特征提取的跨模态一致性检测方法[J]. 图学学报, 2026, 47(2): 286-295.
[15]	赵振兵, 张靖梁, 唐辰康, 毕雨轩, 李浩鹏. 面向积水干扰的变电设备渗漏油精准分割方法[J]. 图学学报, 2026, 47(2): 296-310.